A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).
1, 3, 12, 63, 357, 2112, 12834, 79446, 498504, 3160566, 20202882, 129998400, 841084065, 5466859635, 35672889180, 233564188167, 1533744021741, 10097724827904, 66633102118296, 440600483618184, 2918753549183712, 19367330685385032, 128704927930928088
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..1191
Programs
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Maple
a := n -> if n = 0 then 1 else (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) fi: seq(simplify(a(n)), n = 0..22); # Peter Luschny, Mar 30 2023
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PARI
my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1+x)^2)^(1/3))
Formula
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n+k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (7*n-6)*(n-1)*a(n-1) + 6*(n-2)*a(n-2) - (7*n-22)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * phi^(4*n) / (Gamma(1/3) * 5^(1/6) * n^(2/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 28 2023
a(n) = (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) for n >= 1. - Peter Luschny, Mar 30 2023